Answer Happy

A job-shop has five machines. Machines breakdown at random
following a Poisson process at
the rate of λ per day. Suppose there are three repairmen who can
(individually) fix a broken
machine in half a day on average. Suppose the shop manager receives
an offer to replace the
three repairmen by one “super” repairman who can fix a machine in a
third of the time, but
costs the equivalent of the three repairmen to hire. Answer the
following problems.
1. Model the number of machines failed (and hence in repair) as a
continuous time Markov
chain, in both settings stated above. State clearly what
assumptions you must make to justify
using a Markov model. What is the state space? What is the
transition rate matrix?
2. What queuing models do these two settings correspond to (if
any)?
3. Can you justify hiring the super-repairman? Give a full
mathematical reasoning with
derivation of necessary expressions. (Hint: look at the steady
state mean number of machines
in repair.)